\[ \int \frac {x^2}{\left (b \sqrt {x}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {35 b^{3} \arctanh \left (\frac {\sqrt {a}\, \sqrt {x}}{\sqrt {b \sqrt {x}+a x}}\right )}{4 a^{\frac {9}{2}}}-\frac {4 x^{2}}{a \sqrt {b \sqrt {x}+a x}}+\frac {35 b^{2} \sqrt {b \sqrt {x}+a x}}{4 a^{4}}+\frac {14 x \sqrt {b \sqrt {x}+a x}}{3 a^{2}}-\frac {35 b \sqrt {x}\, \sqrt {b \sqrt {x}+a x}}{6 a^{3}} \]
command
integrate(x^2/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{12} \, \sqrt {a x + b \sqrt {x}} {\left (2 \, \sqrt {x} {\left (\frac {4 \, \sqrt {x}}{a^{2}} - \frac {11 \, b}{a^{3}}\right )} + \frac {57 \, b^{2}}{a^{4}}\right )} + \frac {35 \, b^{3} \log \left ({\left | -2 \, \sqrt {a} {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} - b \right |}\right )}{8 \, a^{\frac {9}{2}}} + \frac {4 \, b^{4}}{{\left (a {\left (\sqrt {a} \sqrt {x} - \sqrt {a x + b \sqrt {x}}\right )} + \sqrt {a} b\right )} a^{4}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________